Bayes by Backprop [1]} can be viewed as a sample-based approximation of Laplace. In order to show this relation, we recall two very powerful equations [2]}. Specifically, let \(\mu \) , \(\Sigma \)  be the parameter of a Gaussian distribution \(q(\theta )\sim \mathcal {N}(\mu ,\Sigma )\) , and let \(V(\theta )\)  be an arbitrary \(L^2\)  integrable function (the log-likelihood in our case). Then, we are interested in the derivatives of \(\mathbb {E}_{\theta \sim q}[V(\theta )]\) . By standard Fourier analysis and integration by part, we have
\(\nabla _\mu \mathbb {E}_{\theta \sim q}[V(\theta )]=\mathbb {E}_{\theta \sim q}[\nabla _\theta V(\theta )],\) 
\(\nabla _\Sigma \mathbb {E}_{\theta \sim q}[V(\theta )]=\frac{1}{2}\mathbb {E}_{\theta \sim q}[\nabla ^2_\theta V(\theta )].\) 
